Retirement and Economic Behavior

By Henry J. Aaron; Gary Burtless | Go to book overview

correlated with X and Z. We deal here with the case in which there are only two relevant points in time. (In our empirical analysis point 1 is 1971 and point 2 is 1977.) Algebraically, the model is

(5) Pr [A(1) = 1] = F [X(1)ʹβ + Z(1)ʹγ + W(1)ʹα]

(6) Pr [A (2) = 1] = F [X(2)ʹβ + Z(2)ʹγ + W(2)ʹα],

where A(i) refers to the variable A measured at point i and where F[·] is the cumulative distribution function of ε(i). That is, the probability of any household living alone in 1971 is a function of the 1971 values of X, Z, and W. By assumption, Z(1) = Z(2) and W(1) = W(2), so that their differences are zero.

Chamberlain shows that when ε follows the logistic distribution,16

(7) Pr(y = 1) = F{βʹ[X(1) - X(2)]},

where

(8)

.

In words, y takes the value 0 if the unit switched from living alone to living with others and 1 if the unit switched from living with others to living alone. Equations 7 and 8 are the usual form of a standard binary logit problem.

Note that respondents for whom A(1) and A(2) have the same value are ignored in equation 7. However, we could have done our analysis using all respondents, whether they changed status or not. But maximizing the likelihood function associated with that problem would yield the same estimates of β as does maximizing the likelihood function associated with equation 7.

As long as X is uncorrelated with ε, the estimates of β will be consistent. Any potential correlation between Z and W and ε is irrelevant since Z and X do not appear. This irrelevance is won at the cost of any estimates of γ and α.


Appendix B: Results from a Cross-Sectional Model of Living Arrangements

In this appendix we present the results from a cross-sectional analysis, despite the potential problems mentioned in appendix A. The operational version of the cross-sectional model is

____________________
16
. "Analysis of Covariance with Qualitative Data."

-248-

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